Stabilisers of eigenvectors of finite reflection groups

Let $x$ be an eigenvector for an element of a finite irreducible reflection group $W$. Let $W_x$ denote the subgroup of $W$ which stabilises $x$. We provide an upper bound for the number of roots in the root system of $W_x$ . This generalises a result of Kostant, who showed that every eigenvector with eigenvalue a primitive $h^\mathrm{th}$ root of unity is regular, where $h$ is the Coxeter number of $W$. We also give a Lie-theoretic interpretation of our result in the study of semisimple conjugacy classes over Laurent series. In a forthcoming paper, we use this result to establish a geometric analogue of a conjecture of Gross and Reeder.